### Archive

Posts Tagged ‘harmonic analysis’

## Areas of Mathematics

For one of my upcoming talks I am trying to include an exhaustive mindmap showing the different areas of Mathematics, and somehow, how they relate to each other. Most of the information I am using has been processed from years of exposure in the field, and a bit of help from Wikipedia.

But I am not entirely happy with what I see: my lack of training in the area of Combinatorics results in a rather dry treatment of that part of the mindmap, for example. I am afraid that the same could be told about other parts of the diagram. Any help from the reader to clarify and polish this information will be very much appreciated.

And as a bonus, I included a $\LaTeX$ script to generate the diagram with the aid of the tikz libraries.

\tikzstyle{level 2 concept}+=[sibling angle=40]
\begin{tikzpicture}[scale=0.49, transform shape]
\path[mindmap,concept color=black,text=white]
node[concept] {Pure Mathematics} [clockwise from=45]
child[concept color=DeepSkyBlue4]{
node[concept] {Analysis} [clockwise from=180]
child {
node[concept] {Multivariate \& Vector Calculus}
[clockwise from=120]
child {node[concept] {ODEs}}}
child { node[concept] {Functional Analysis}}
child { node[concept] {Measure Theory}}
child { node[concept] {Calculus of Variations}}
child { node[concept] {Harmonic Analysis}}
child { node[concept] {Complex Analysis}}
child { node[concept] {Stochastic Analysis}}
child { node[concept] {Geometric Analysis}
[clockwise from=-40]
child {node[concept] {PDEs}}}}
child[concept color=black!50!green, grow=-40]{
node[concept] {Combinatorics} [clockwise from=10]
child {node[concept] {Enumerative}}
child {node[concept] {Extremal}}
child {node[concept] {Graph Theory}}}
child[concept color=black!25!red, grow=-90]{
node[concept] {Geometry} [clockwise from=-30]
child {node[concept] {Convex Geometry}}
child {node[concept] {Differential Geometry}}
child {node[concept] {Manifolds}}
child {node[concept,color=black!50!green!50!red,text=white] {Discrete Geometry}}
child {
node[concept] {Topology} [clockwise from=-150]
child {node [concept,color=black!25!red!50!brown,text=white]
{Algebraic Topology}}}}
child[concept color=brown,grow=140]{
node[concept] {Algebra} [counterclockwise from=70]
child {node[concept] {Elementary}}
child {node[concept] {Number Theory}}
child {node[concept] {Abstract} [clockwise from=180]
child {node[concept,color=red!25!brown,text=white] {Algebraic Geometry}}}
child {node[concept] {Linear}}}
node[extra concept,concept color=black] at (200:5) {Applied Mathematics}
child[grow=145,concept color=black!50!yellow] {
node[concept] {Probability} [clockwise from=180]
child {node[concept] {Stochastic Processes}}}
child[grow=175,concept color=black!50!yellow] {node[concept] {Statistics}}
child[grow=205,concept color=black!50!yellow] {node[concept] {Numerical Analysis}}
child[grow=235,concept color=black!50!yellow] {node[concept] {Symbolic Computation}};
\end{tikzpicture}


## So you want to be an Applied Mathematician

The way of the Applied Mathematician is one full of challenging and interesting problems. We thrive by association with the Pure Mathematician, and at the same time with the no-nonsense, hands-in, hard-core Engineer. But not everything is happy in Applied Mathematician land: every now and then, we receive the disregard of other professionals that mistake either our background, or our efficiency at attacking real-life problems.

I heard from a colleague (an Algebrist) complains that Applied Mathematicians did nothing but code solutions of partial differential equations in Fortran—his skewed view came up after a naïve observation of a few graduate students working on a project. The truth could not be further from this claim: we do indeed occasionally solve PDEs in Fortran—I give you that—and we are not ashamed to admit it. But before that job has to be addressed, we have gone through a great deal of thinking on how to better code this simple problem. And you would not believe the huge amount of deep Mathematics that are involved in this journey: everything from high-level Linear Algebra, Calculus of Variations, Harmonic Analysis, Differential Geometry, Microlocal Analysis, Functional Analysis, Dynamical Systems, the Theory of Distributions, etc. Not only are we familiar with the basic background on all those fields, but also we are supposed to be able to perform serious research on any of them at a given time.

My soon-to-be-converted Algebrist friend challenged me—not without a hint of smugness in his voice—to illustrate what was my last project at that time. This was one revolving around the idea of frames (think of it as redundant bases if you please), and needed proving a couple of inequalities involving sequences of functions in $L_p$—spaces, which we attacked using a beautiful technique: Bellman functions. About ninety minutes later he conceded defeat in front of the board where the math was displayed. He promptly admitted that this was no Fortran code, and showed a newfound respect and reverence for the trade.

It doesn’t hurt either that the kind of problems that we attack are more likely to attract funding. And collaboration. And to be noticed in the press.

Alright, so some of you are sold already. What is the next step? I am assuming that at his point you own your Calculus, Analysis, Probability and Statistics, Linear Programming, Topology, Geometry, Physics and you are able to solve most known ODEs. From here, as with any other field, my recommendation is to slowly build a Batman belt: acquire and devour a sequence of books and scientific articles, until you are very familiar with their contents. When facing a new problem, you should be able to recall from your Batman belt what technique could work best, in which book(s) you could get some references, and how it has been used in the past for related problems.

Following these lines, I have included below an interesting collection with the absolutely essential books that, in my opinion, every Applied Mathematician should start studying:

## The hunt for a Bellman Function.

September 26, 2008 1 comment

This is a beautiful and powerful mathematical technique in Harmonic Analysis that allows, among other things, to prove very complicated inequalities in the theory of Singular Integral Operators, without using much of the classical machinery in this field.

The Bellman function was the tool that allowed their creators (Fedor Nazarov and Sergei Treil) to crack the problem of weighted norm inequalities with matrix weights for the case $\boldsymbol{p} \neq \boldsymbol{2}$ and finally solve it completely.

Copies of the original paper can be found at the authors’ pages; e.g. [www.math.brown.edu/~treil/papers/bellman/bell3.ps] (notice the postscript file is huge, as the article has more than 100 pages).

Let me illustrate the use of Bellman functions to solve a simple problem:

Dyadic-$\boldsymbol{L}_\mathbf{2}(\mathbb{R})$ version of the Carleson Imbedding Theorem

Let $\mathcal{D}$ be the set of all dyadic intervals of the real line. Given a function $f \in L_1^{\text{loc}}(\mathbb{R})$, consider the averages $\langle f \rangle_I = \lvert I\rvert^{-1} \int_I f$, on each dyadic interval $I \in \mathcal{D}$.
Let $\{ \mu_I \geq 0 \colon I \in \mathcal{D} \}$ be a family of non-negative real values satisfying the Carleson measure condition—that is, for any dyadic interval $I \in \mathcal{D}$, $\sum_{J \subset I, J~\text{dyadic}} \mu_J \leq \lvert I \rvert.$
Then, there is a constant $C>0$ such that for any $f \in L_2(\mathbb{R})$,

$\displaystyle{\sum_{ I \in \mathcal{D} } \mu_I \lvert \langle f \rangle_{I} \rvert^2 \leq C \lVert f \rVert_{L_2(\mathbb{R})}^2}$

Categories: Analysis

## Presentation: Hilbert Transform Pairs of Wavelets

In the first IMI seminar, I presented an introduction to the survey paper “The Dual-Tree Complex Wavelet Transform“, by Selesnick, Baraniuk and Kingsbury. It was meant to be a (very) basic overview of the usual techniques of signal processing with an emphasis on wavelet coding, an exposition on the shortcomings of real-valued wavelets that affect the work we do at the IMI, and the solutions proposed by the three previous authors. In a subsequent talk, I will give a more mathematical (and more detailed) account on filter design for the dual-tree $\mathbb{C}$WT. Click on the image below to retrieve a pdf version of the presentation.